If you cannot get to the factors, or find one for £40 on ebay try taking the battery inside overnight.
Do you access to a trickle charger? or in fact any charger
Get a bump start short term, get battery out of car and charge it .
Or , Get a bump start drive around for a while to charge then keep it next to a rad indoors overnight
However , if its a loud whirring it still might be starter motor failing to engage the ring gear. Solenoids tend to clickety clack or do nothing as a solenoid is a powered switch. The starter might be pre -engaged so at engine off the cogs are meshed together, then on start up the cogs disengage
Or the strter motor throws a cog up a rifled spindle to engage with the teeth on the flywheel/.If this is damage or the rifling full of crap then a whirrring is all you get.
Local tyre fitters will have a batttery tester , grab some biccys and ask them to run the tester over over battery for amps, not volts.